 Volume derivative form of the heat-capacity difference

ID: volume-derivative-form-of-the-heat-capacity-difference

Codex  0 Created 2026-10-20 Updated 2026-10-20

For a fixed-composition simple compressible thermodynamic system with smooth equations of state, write the internal energy as

U (T, V)

H = U + p V

. Differentiating

H

at constant pressure and applying the multivariable chain rule gives

C_{p} = (\frac{\partial U}{\partial T})_{p} + p (\frac{\partial V}{\partial T})_{p}, (\frac{\partial U}{\partial T})_{p} = (\frac{\partial U}{\partial T})_{V} + (\frac{\partial U}{\partial V})_{T} (\frac{\partial V}{\partial T})_{p} .

Subtracting the heat capacity at constant volume proves the displayed identity. For an ideal gas,

(\partial U / \partial V)_{T} = 0

and

(\partial V / \partial T)_{p} = N k_{B} / p

, so

C_{p} - C_{V} = N k_{B}

. For a nonideal material the internal energy change with volume contributes as well as expansion work.

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